The statistical interpretation according to Born and

Heisenberg

Guido Bacciagaluppi∗

Abstract

At the 1927 Solvay conference Born and Heisenberg presented ajoint report on quantum mechanics. I suggest that the significance ofthis report lies in that it contains a ‘final’ formulation of the statisticalinterpretation of quantum mechanics that goes beyond Born’s origi-nal proposal. In particular, this formulation imports elements fromHeisenberg’s work as well as from the transformation theory of Diracand Jordan. I suggest further a reading of Born and Heisenberg’sposition in which the wave function is an effective notion. This canmake sense of a remarkable aspect of their presentation, namely thefact that the ‘quantum mechanics’ of Born and Heisenberg apparentlylacks wave function collapse.

1 Introduction

The fifth Solvay conference of 1927 saw the presentation of (and confronta-tion between) three fundamental approaches to quantum theory: de Broglie’spilot-wave theory, Schrodinger’s wave mechanics, and ‘quantum mechanics’(i.e. matrix mechanics and its further developments), the latter presentedto the conference in a joint report by Born and Heisenberg.

A thorough examination of the conference proceedings reveals substantialamounts of material that are either little known or generally misrepresented.Such an examination is given in a forthcoming book on the 1927 Solvay con-ference (Bacciagaluppi and Valentini, 2008), which also includes a complete

∗Centre for Time, Department of Philosophy, University of Sydney, Sydney NSW 2006,Australia (e-mail: guido.bacciagaluppi@arts.usyd.edu.au).

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English version of the proceedings, based on the original-language materialswhere available.1

In this paper, I wish to focus on the report by Born and Heisenberg, ar-guing that it contains a version of the statistical interpretation of quantummechanics that goes well beyond that elaborated by Born in his papers oncollisions and in his paper on the adiabatic theorem (Born, 1926a,b,c). Inparticular, the report offers an interpretation of the interference of proba-bilities, which appears to be related to Heisenberg’s ideas as developed inhis uncertainty paper (Heisenberg, 1927).

I shall further propose a reading of Born and Heisenberg’s position in whichthe wave function has no fundamental status, in a way related to Heisen-berg’s paper on fluctuations (Heisenberg, 1926). Born and Heisenberg’s re-port should thus indeed be seen as presenting an approach that is fundamen-tally different from both de Broglie’s pilot-wave theory and Schrodinger’swave mechanics.

Finally, I suggest that the proposed reading makes sense of an aspect of Bornand Heisenberg’s presentation (and of the discussions) that is especiallypuzzling from the point of view of a modern reader, namely the almosttotal absence of the ‘collapse of the wave function’ or ‘reduction of the wavepacket’.

Much of the material presented below is based on Bacciagaluppi and Valen-tini (2008),2 including parts of the book that are joint work or even princi-pally the work of my coauthor (the latter especially in section 3). However,the perspectives on this material adopted in the paper and in the book

1Quotations below from the proceedings of the conference are based on this Englishedition; page references are to the corresponding passages of the on-line draft available athttp://xxx.arxiv.org/abs/quant-ph/0609184 .

2The report itself and the discussion following it are translated and annotated onpp. 408–447. Born and Heisenberg’s views are analysed and discussed principally in chap-ters 3 and 6. Among the topics discussed in this paper, the main ones treated in thebook are the following. Born and Heisenberg’s treatment of interference is discussed insection 6.1.2 (pp. 172–177). The derivation of transition probabilities in Born’s collisionpapers and in Heisenberg’s fluctuations paper are discussed, respectively, in section 3.4.3(pp. 107–108) and 3.4.4 (pp. 109–111). Phase randomisation in measurement is discussedin detail on pp. 173–177. Extensive presentations and analyses of Born’s discussion ofthe cloud chamber and of the exchange between Heisenberg and Dirac are given, respec-tively, in sections 6.2 (pp. 177-182) and 6.3 (pp. 182–189). Finally, Einstein’s alternativehidden-variables proposal (with Heisenberg’s comments) is discussed in detail in section11.3 (pp. 259–265).

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are very different. The idea of a ‘definitive’ version of the statistical in-terpretation merging elements from Born’s and Heisenberg’s work is hardlymentioned in the book. Furthermore, the presentation in the book is uncom-mittal about the views on collapse and on the status of the wave functionheld by Born and Heisenberg. This paper instead attempts to put forwardone particular reading (not because it is unequivocally supported by theevidence, but as a proposal for making sense of the material that will needfurther evaluation).

It is useful therefore to spell out at least some of the differences between thetreatment of the material in this paper and in Bacciagaluppi and Valentini(2008). First of all, as emphasised already, here I suggest that the report is anew stage of development of the statistical interpretation. This is somethingthat is left largely implicit in the discussion in the book. Here I suggest thatBorn and Heisenberg present a single coherent position. The treatment inthe book allows for possible differences in opinion between the two authors(emphasising for instance the possible relation between Born’s discussion ofthe cloud chamber and the guiding-field ideas in his collision papers). HereI try to make explicit links between Born and Heisenberg’s implicit notionof state in their treatment of transition probabilities on the one hand, andBorn’s treatment of the cloud chamber on the other; I also hint at thepossibility that Pauli had such a link in mind. Neither suggestion is madein the book. Last but not least, I suggest here that Born and Heisenbergdid not believe in the reality of the wave function. This is mentioned in thebook only as one tentative possibility among others.

Bacciagaluppi and Valentini (2008) refrains on purpose from drawing con-clusions from the material that might have been premature. This paperhopes to be a first step in drawing further conclusions. Indeed, while theinterpretation of quantum theory seems as highly controversial again todayas it was in 1927, from the vantage point of eighty years of philosophy ofquantum physics a more dispassionate evaluation of the sources in the inter-pretation debate should be possible. I wish to thank Antony Valentini fordiscussion and comments during the preparation of this paper, although ofcourse all deviations from and additions to the presentation of the materialas given in the book are my sole responsibility.

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2 The statistical interpretation in Born and Heisen-berg’s report

The report by Born and Heisenberg on ‘quantum mechanics’ is surprisinglydifficult for the modern reader. This is partly because Born and Heisenbergare describing various stages of development of the theory that are quitedifferent from today’s quantum mechanics. At the same time, the interpre-tation of the theory also appears to have undergone important modifications,in particular regarding the notion of the state of a system (see Bacciagaluppiand Valentini, 2008, section 3.4).

It is known which sections of the report were drafted by Born and whichby Heisenberg. In particular, the section most relevant to our concerns —that on the ‘Physical interpretation’ of the theory — was drafted by Born,who also prepared the final version of the paper, although Heisenberg madesome further small changes.3 As we shall see, the interpretation presentedmerges crucially elements of Born’s and Heisenberg’s work, and (at least forthe purposes of this paper) we shall consider the interpretational views asset forth in the report (and in the discussions reported below) as express-ing a common voice. This is also supported by Born’s remark to Lorentzthat Heisenberg and he were ‘of one and the same opinion on all essentialquestions’.4

2.1 The statistical interpretation

Until the 1927 report, the most explicit presentation of the statistical inter-pretation of quantum theory was that given in Born’s paper on the adiabatictheorem (1926c). The picture presented by Born is as follows. Particles ex-ist, at least during periods in which systems evolve freely (say, between 0and t). At the same time, they are accompanied by de Broglie-Schrodingerwaves ψ. Regardless of the form of these waves, during a period of freeevolution a system is always in a stationary state. When the waves ψ aredeveloped in the basis of eigenstates ψn(x) of energy, say

ψ(x, 0) =∑

n

cnψn(x) , (1)

3Born to Lorentz, 29 August 1927, AHQP-LTZ-11 (in German). Cf. Bacciagaluppiand Valentini (2008, section 3.2).

4Born to Lorenz, loc. cit.; quoted with the kind permission of Prof. Gustav Born.

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they yield the probabilities for the occurrence of the stationary states, the‘state probabilities’ being given by |cn|2. During periods, say from t to T ,in which an external force is applied (or the system interacts with anothersystem) there may be no anschaulich representation of the processes takingplace. As regards the particles, the only thing that can be said is that‘quantum jumps’ occur, in that after the external influence has ceased thesystem is generally in a different stationary state. The evolution of thestate probabilities instead is well-defined and determined by the Schrodingerequation, in the sense that the state probabilities at time T are given by thecorresponding expression |Cn|2 of the coefficients of ψ(x, T ).

For the case in which ψ(x, 0) = ψn(x), Born determines explicitly thesecoefficients, call them bnm, in terms of the time-dependent external potential;thus,

ψ(x, T ) =∑m

bnmψm(x) . (2)

Given the interpretation of the quantities |bnm|2 as state probabilities, inthis case they are also the ‘transition probabilities’ for the jump from theinitial state, which by assumption is ψn(x) at time t, to the final state ψm(x)at time T .

Finally, for the general case of an initial superposition (1), Born states thatthe state probabilities |Cn|2 have the form

|Cn|2 = |∑m

cmbmn|2 , (3)

noting that (1926c, p. 174):

The quantum jumps between two states labelled by m andn thus do not occur as independent events; for in that case theabove expression should be simply

∑m |cm|2|bmn|2

(with a footnote to Dirac (1926) as also pointing out this fact5). He alsoremarks that, as he will show later on, the quantum jumps become indepen-dent in the case of an external perturbation by “‘natural” light’.

As it appears in Born’s adiabatic paper, the statistical interpretation isquite different both from the familiar textbook interpretations and from the

5Cf. especially pp. 674 and 677 of Dirac’s paper.

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interpretation we shall find in Born and Heisenberg’s Solvay report. Forinstance, the requirement that the state of an isolated system be alwaysa stationary state is unfamiliar, to say the least. (As we shall see, it iseventually relaxed in Born and Heisenberg’s report.)

For now let us focus on Born’s remark about quantum jumps not beingindependent. This terminology appears to presuppose a probability space inwhich the elementary events do not correspond to single systems performingquantum jumps, but to N -tuples of systems all performing quantum jumpsbetween t and T .6 (The analogous case in classical statistical mechanics isthe treatment of gases of interacting rather than non-interacting particles.)

If this is the correct way of understanding Born’s statistical interpretation ofthe wave function (at least as proposed in 1926), then Einstein may well havehad Born’s view in mind when at the 1927 Solvay conference he criticisedwhat he labelled ‘conception I’ of the wave function (p. 487):7

The de Broglie-Schrodinger waves do not correspond to a singleelectron, but to a cloud of electrons extended in space. Thetheory gives no information about individual processes, but onlyabout the ensemble of an infinity of elementary processes.

According to Einstein, it is only the alternative ‘conception II’, in which thewave function is a complete description of an individual system (and whichhe also goes on to criticise), that enables one to derive the conservationlaws, the results of the Bothe-Geiger experiments and the straight tracks ofα-particles in a cloud chamber. Note that the last example is taken up byBorn in the general discussion (see below section 3.2).

Be it as it may, Born’s paper on the adiabatic theorem lacks a separatediscussion of interference; and this is the crucial point where the report byBorn and Heisenberg goes further than Born’s paper. Born and Heisen-berg (p. 423) consider an atom that is initially in a superposition of energystates ψn(x), with coefficients cn(0) = |cn(0)| eiγn and eigenvalues En. The

6Born’s discussion of natural light later in the paper only reinforces this impression.Born assumes that due to the irregular temporal course of the external perturbation, thebnm will fluctuate independently.

7For an alternative interpretation of Einstein’s comments, see Bacciagaluppi and Valen-tini (2008, p. 225).

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Schrodinger equation induces a time evolution

cn(t) =∑m

Snm(t)cm(0) . (4)

In the special case where cm(0) = δmk for some k, we have |cn(t)|2 =|Snk(t)|2, and Born and Heisenberg interpret |Snk(t)|2 as a transition prob-ability. They also draw the conclusion that ‘the |cn(t)|2 must be the stateprobabilities’ (p. 424). Thus far the discussion is reminiscent of Born’s treat-ment, and Born and Heisenberg in fact quote Born’s paper on the adiabaticprinciple in support of this interpretation.

At this point, however, Born and Heisenberg recognise a ‘difficulty of prin-ciple’ (p. 424), which is precisely that for an initial superposition of energystates the final probability distribution is given by

|cn(t)|2 =∣∣∣ ∑

m

Snm(t)cm(0)∣∣∣2 , (5)

as opposed to|cn(t)|2 =

∑m

|Snm(t)|2 |cm(0)|2 . (6)

This ‘theorem of the interference of probabilities’ in Born and Heisenberg’swords appears to contradict what ‘one might suppose from the usual prob-ability calculus’ (p. 424).

Born and Heisenberg then make a remarkable statement (pp. 424–425):

.... it should be noted that this ‘interference’ does not rep-resent a contradiction with the rules of the probability calculus,that is, with the assumption that the |Snk|2 are quite usual prob-abilities. In fact, .... [(6)] follows from the concept of probability.... when and only when the relative number, that is, the proba-bility |cn|2 of the atoms in the state n, has been established be-forehand experimentally. In this case the phases γn are unknownin principle, so that [(5)] then naturally goes over to [(6)] .... .

We shall return in the next section to Born and Heisenberg’s characterisationof the role of the experiment. What they are saying about the probabilitycalculus is that the expressions |Snk|2 denote ‘usual’ transition probabilitiesirrespectively of whether they appear in (5) or in (6). Instead, the reason

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for the failure of (6) to hold in general is that the expressions |cm|2 are notalways state probabilities, because the state probabilities themselves are notalways well-defined (Bacciagaluppi and Valentini, 2008, pp. 175–176). If thestate probabilities are well-defined (namely if the energy has been measured,in general non-selectively), then one can calculate them at future times using(6). The truth of this conditional statement, however, is not affected if thestate probabilities in fact are not always well-defined.

This, now, is analogous to Heisenberg’s famous discussion of the ‘law ofcausality’ in his uncertainty paper: the law is again a conditional statement,which remains true although the state of the system is defined in fact onlyto within the accuracy given by the uncertainty principle. In Heisenberg’sown words: ‘.... in the sharp formulation of the law of causality, “If we knowthe present exactly, we can calculate the future”, it is not the consequentthat is wrong, but the antecedent. We cannot in principle get to know thepresent in all determining data’ (Heisenberg, 1927, p. 197).8

What Born and Heisenberg mean by ‘usual’ transition probabilities is evi-dently not the idea of conditional probabilities defined as quotients of theabsolute probabilities, since for them the latter are not always well-defined.Instead they must mean some kind of potentialities, some probabilistic ‘fieldof force’, existing independently of the presence of a ‘test particle’.

Regarding the ‘state’ of the system, the picture they have in mind seemsto be similar to that in Born’s papers: namely, that the actual state of theatom is a state of definite energy. The difference to the earlier picture isthat now the stationary states exist or have a well-defined distribution onlyupon measurement (although the question of why this should be so is notexplicitly addressed). Instead, the wave function merely defines a statisticaldistribution over the stationary states.

The step to considering arbitrary observables, and not just the energy, ashaving definite values only upon measurement is now very easy.9 In orderto extend the above picture to the general case, one has to generalise Bornand Heisenberg’s notion of transition probability to the case in which twodifferent observables are measured at the beginning and the end of a giventime interval. Here Born and Heisenberg are not very explicit. What they

8On Heisenbergs treatment of the ‘law of causality’, see also Beller (1999, pp. 110–113).9Again, Heisenberg’s uncertainty paper (Heisenberg, 1927, pp. 190–191), as well as

his correspondence with Pauli (Heisenberg to Pauli, 23 February 1927, in Pauli, 1979, pp.376–382) both mention explicitly the loss of a privileged status for stationary states.

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actually do in the report is to define ‘relative state probabilities’, i.e. equal-time conditional probabilities for values of one quantity given the value ofanother, in terms of the projections of the eigenvectors (‘principal axes’) ofone quantity onto the eigenvectors of the other. (In modern terminology,it is of course these expressions that are called ‘transition probabilities’.)In this they follow Dirac’s (1927) and Jordan’s (1927b,c) development ofthe transformation theory, which Heisenberg understood as generalising theideas of his paper on fluctuations (Heisenberg, 1926).10

2.2 Transition probabilities and the status of the wave func-tion

In Born’s work as presented above, the statistical interpretation is an in-terpretation of Schrodinger’s theory, albeit ‘in Heisenberg’s sense’ (Born,1926c, p. 168). As we shall see now, instead, Born and Heisenberg in thereport do not start directly with the Schrodinger equation. I shall suggestthat in Born and Heisenberg’s view, although they may be very useful toolsboth for calculational purposes and for understanding interference, the wavefunction and the Schrodinger equation are only effective notions.

Section II of the report, on the ‘physical interpretation’ of quantum mechan-ics, begins with the following statement (p. 420):

The most noticeable defect of the original matrix mechanics con-sists in that at first it appears to give information not about ac-tual phenomena, but rather only about possible states and pro-cesses. It allows one to calculate the possible stationary states ofa system; further it makes a statement about the nature of theharmonic oscillation that can manifest itself as a light wave ina quantum jump. But it says nothing about when a given stateis present, or when a change is to be expected. The reason forthis is clear: matrix mechanics deals only with closed periodicsystems, and in these there are indeed no changes. In order tohave true processes, as long as one remains in the domain of ma-trix mechanics, one must direct one’s attention to a part of the

10Heisenberg to Pauli, 23 November 1926: ‘Here [in Copenhagen] we have also beenthinking more about the question of the meaning of the transformation function S andDirac has achieved an extraordinarily broad generalisation of this assumption from mynote on fluctuations’ (in Pauli, 1979, p. 357).

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system; this is no longer closed and enters into interaction withthe rest of the system. The question is what matrix mechanicscan tell us about this.

As raised here, the question to be addressed is how to incorporate intomatrix mechanics the (actual) state of a system, and the time developmentof such a state.

Two methods for introducing change into matrix mechanics are then pre-sented. First of all, following Heisenberg’s paper on fluctuation phenomena(Heisenberg, 1926), Born and Heisenberg consider the matrix mechanical de-scription of two coupled systems in resonance. This they interpret in termsof quantum jumps between the energy levels of the two systems, and theygive an explicit expression for the corresponding transition probabilities. Itis only after this matrix mechanical discussion that Born and Heisenbergintroduce the time-dependent Schrodinger equation as a way for describingtime dependence. From this, Born and Heisenberg then derive transitionprobabilities following Born’s adiabatic paper (1926c), as described above.

Already in the collision papers Born had aimed precisely at including intomatrix mechanics a description of the transitions between stationary states(Born, 1926a,b). Born had managed to describe the asymptotic behaviour ofthe combined system of electron and atom solving by perturbation methodsthe time-independent Schrodinger equation, yielding a superposition of com-ponents associated to various, generally inelastic, collisions in which energyis conserved. Interpreting statistically the coefficients in the expansion, andsince the incoming asymptotic wave function corresponds to a fully deter-mined stationary state and ‘uniform rectilinear motion’,11 one obtains theprobabilities for quantum jumps from the given ‘initial’ state to the given‘final’ state, i.e. the desired transition probabilities.12

At first Born may have thought that wave mechanical methods were indis-pensable for this purpose.13 To Heisenberg’s delight, however, Pauli was

11This is, indeed, Born’s terminology (1926a, p. 864; 1926b, p. 806). In this context,cf. also the discussion of Born and Wiener (1926) in Bacciagaluppi and Valentini (2008,section 3.4.1).

12Note that Born considers indeed two conceptually distinct objects: on the one handthe stationary states of the atom and the electron, on the other hand the wave functionthat defines the probability distribution over the stationary states. He reserves the word‘state’ only for the stationary states.

13Cf. Born to Schrodinger, 16 May 1927: ‘the simple possibility of treating with it

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able to sketch how one could reinterpret Born’s results in terms of matrixelements.14 A few days later, Heisenberg sent Pauli the manuscript of hispaper on fluctuation phenomena (Heisenberg, 1926), in which he developedconsiderations similar to Pauli’s ones in the context of the example of twoatoms in resonance. Indeed, starting from a closed system (thus stationaryfrom the point of view of matrix mechanics) and focussing on the descriptionof the subsystems, Heisenberg was able to derive explicit expressions for thetransition probabilities within matrix mechanics proper, without having tointroduce the wave function as an external aid. A very similar result wasderived at the same time by Jordan (1927a), using two systems with a singleenergy difference in common.

Born’s collision papers and the papers by Heisenberg and by Jordan can beall understood as seeking to obtain ‘information .... about actual phenom-ena’, by ‘direct[ing] one’s attention to a part of the system’. In this context,the fact that it is Heisenberg’s setting rather than Born’s which is chosenin the report suggests that Born and Heisenberg indeed intend to make thepoint that matrix mechanics can account for time-dependent phenomenawithout the aid of wave mechanics.

It is in this sense, I suggest, that one should read the following remark madeby Born and Heisenberg between their introduction of the time-dependentSchrodinger equation and their discussion of transition probabilities andinterference (p. 423):

Essentially, the introduction of time as a numerical variable re-duces to thinking of the system under consideration as coupledto another one and neglecting the reaction on the latter, but thisformalism is very convenient and leads to a further developmentof the statistical view.

In particular, I suggest that in Born and Heisenberg’s view one shouldnot simply interpret a time-dependent external potential in the Schrodingerequation (as used in the adiabatic paper for instance) as a substitute for the

aperiodic processes (collisions) made me first believe that your conception was superior’(quoted in Mehra and Rechenberg, 2000, p. 135).

14See Pauli to Heisenberg, 19 October 1926, in Pauli (1979, pp. 340–349), and Heisen-berg’s reply: ‘Your calculations have given me again great hope, because they show thatBorn’s somewhat dogmatic viewpoint of the probability waves is only one of many possibleschemes’ (Heisenberg to Pauli, 28 October 1926, in Pauli, 1979, p. 350).

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full Schrodinger equation of the combined system, but that the Schrodingerequation itself arises from considering only subsystems.15

This reading is further supported by Born and Heisenberg’s remarks ongeneralising transition probabilities to the case of an arbitrary observable,which are now coached in terms that bypass wave functions entirely (pp. 428–429):

Alongside the concept of the relative state probability |ϕ(q′, Q′)|2,there also occurs the concept of transition probability, namely,whenever one considers a system as depending on an externalparameter, be it time or any property of a weakly coupled exter-nal system. Then the system of principal axes of any quantitybecomes dependent on this parameter; it experiences a rotation,represented by an orthogonal transformation S(q′, q′′), in whichthe parameter enters .... . The quantities |S(q′, q′′)|2 are the‘transition probabilities’; in general, however, they are not in-dependent, instead the ‘transition amplitudes’ are composed ac-cording to the interference rule.

In part, reference to wave functions here is eliminated through a switch to theHeisenberg picture. One should note, however, that Born and Heisenbergmanage to eliminate reference to the wave function completely only becausethey consider exclusively maximal observables. In the more general caseof non-maximal (i.e. coarse-grained) observables,16 transition probabilities(whether in their sense or in the modern sense) depend also on the quantumstate.

The overall picture one glimpses from these aspects of Born and Heisenberg’sremarks is that what exists are just transition probabilities and measuredvalues (although, as mentioned already, it is not explained why measurementshould play such a special role).

As regards the transition probabilities, the |Snk|2 defined by Born andHeisenberg are independent of the actual wave function. They can be cal-culated using the formalism of wave functions, namely as the coefficients in

15Cf. also the derivation of time-dependent transition probabilities in Heisenberg (1930,pp. 148–150).

16And of course in the most general case of observables as positive-operator-valuedmeasures (POVMs), for which see e.g. Peres (1993, pp. 282–289).

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(4) for the case in which the initial wave function is the kth eigenstate ofenergy, but they are taken as the correct transition probabilities even whenthe initial wave function is arbitrary.

By way of contrast, one could take Bell’s (1987) discrete and stochasticversion of de Broglie’s pilot-wave theory. In a theory of this type, givena choice of preferred observable (‘beable’ in Bell’s terminology), the |cn|2are indeed always state probabilities, and one constructs appropriate tran-sition probabilities that are generally different from Born and Heisenberg’s|Snk|2, thereby explicitly retaining the validity of the standard formula (6).Evidently, Bell’s transition probabilities must depend on the actual wavefunction of the system, which thus acts as a pilot wave, as in de Broglie’stheory. Born and Heisenberg instead choose to give up the |cn|2 as stateprobabilities and to keep the transition probabilities independent of the ac-tual wave function (which is thus not a pilot wave in any sense).

In general, wave functions themselves can usefully represent statistical in-formation about measured values, but one need not consider wave functionsas describing the real state of the system (contra Schrodinger). In this sense,they appear to resemble more the Liouville distributions of classical mechan-ics, a comparison suggested also by some of Born and Heisenberg’s remarks(p. 433):17

For some simple mechanical systems .... the quantum mechan-ical spreading of the wave packet agrees with the spreading ofthe system trajectories that would occur in the classical theoryif the initial conditions were known only with the precision re-striction [given by the uncertainty principle]. .... But in generalthe statistical laws of the spreading of a ‘packet’ for the classicaland the quantum theory are different ....

As Darrigol (1992, p. 344) has emphasised, there is no notion of state vectoreither in Dirac’s paper on the transformation theory (Dirac, 1927). (Thewell-known bras and kets do not appear yet.) The main result of Dirac’spaper is to determine the conditional probability density for one observablegiven a value for a different observable, a result that Dirac illustrates bydiscussing precisely Heisenberg’s example of transition probabilities in res-onant atoms and Born’s collision problem. As we shall see in section 3.3,

17Note also that in his discussion of the cloud chamber, Born once refers to the wavepacket as a ‘probability packet’ (p. 483).

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however, by the time of the Solvay conference Dirac’s and Born and Heisen-berg’s views had diverged, both with regard to whether the wave functionshould describe ‘the state of the world’, and with regard to the notion of thecollapse of the wave function.

3 Measurements and effective collapse

It is remarkable that the reduction of the wave packet is totally absent fromBorn and Heisenberg’s report, although this concept had been famously in-troduced by Heisenberg himself in the uncertainty paper (Heisenberg, 1927,p. 186). In this section we shall discuss what appears to take the place ofreduction in Born and Heisenberg’s report, then we shall focus on the twoplaces in the conference proceedings where the reduction of the wave packetappears explicitly: Born’s treatment of the cloud chamber in his main discus-sion contribution (pp. 483–486) and the intriguing exchange between Diracand Heisenberg (pp. 494–497), both appearing in the general discussion atthe end of the conference.

3.1 Measurement and phase randomisation

What is Born and Heisenberg’s description of measurement? In the report,measurement appears only in the discussion of interference, namely, as wehave seen, as the source for its suppression. This suppression of interferenceis achieved neither by applying the ‘reduction of the wave packet’ (i.e. notby collapsing the wave function onto the eigenstates of the measured observ-able) nor through entanglement of the measured system with the measuringapparatus (a simple form of what we would now call decoherence). Thelatter would in fact presuppose a quantum mechanical treatment of the in-teraction between the two, which was uncharacteristic for the time.

Instead, Born and Heisenberg appear to take measurement as introducing arandomisation of the phase in the wave function (Bacciagaluppi and Valen-tini, 2008, p. 173–177): indeed, they consider the case in which (p. 425):

.... the relative number, that is, the probability |cn|2 of theatoms in the state n, has been established beforehand experi-mentally. In this case the phases γn are unknown in principle,

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so that [(5)] then naturally goes over to [(6)] .... .

At this point Born and Heisenberg add a reference to Heisenberg’s uncer-tainty paper, which indeed contains a more detailed version of essentiallythe same claim (see also below section 3.3). There, Heisenberg considers aStern-Gerlach atomic beam passing through two successive regions of fieldinhomogeneous in the direction of the beam (so as to induce transitions be-tween energy states without separating the beam into components). If theinput beam is in a definite energy state then the beam emerging from thefirst region will be in a superposition. The probability distribution for energyemerging from the second region will then contain interference — as in (5),where the ‘initial’ superposition (1) is now the state emerging from the firstregion. Heisenberg asserts that, if the energy of an atom is actually mea-sured between the two regions, then because of the resulting perturbation‘the “phase” of the atom changes by amounts that are in principle uncon-trollable’ (Heisenberg, 1927, pp. 183–184), and averaging over the unknownphases in the final superposition yields a non-interfering result.

This is clearly not the same as applying the collapse postulate. Indeed, ifone applied the usual ‘Dirac-von Neumann’ postulate, after the measurementthe atoms would be in eigenstates of energy, and the non-interfering resultwould be obtained by averaging over the different energy values.

The difference between the two descriptions is masked by the fact that theaverages are the same, i.e. a statistical mixture of states of the superposedform (1), with randomly-distributed phases γn, is indeed statistically equiv-alent to a mixture of energy states ψn(x) with weights |cn(0)|2, because thecorresponding density operators are the same. But for the subensemblesselected on the basis of the measurement results (i.e. for the subensem-bles with definite values for the energy), the density operators are clearlydifferent.

In the standard collapse case, indeed, the selected subensemble is homo-geneus and described by a pure state ψn(x). In the case of phase ran-domisation, taken literally, the subensembles selected on the basis of themeasurement results are instead described by the same mixture of super-posed states with randomly-distributed phases γn. If we take the state ofthe system (in the modern sense, i.e. the density operator) as determin-ing the probabilities for the results of future measurements, we ought toconclude that in the case of phase randomisation an immediate repetition

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of the measurement will generally not yield the same result as the originalmeasurement, and that any value could occur as a possible result.

However, if our reading above of Born and Heisenberg’s discussion of theprobability calculus is correct, the quantum state in the modern sense is notwhat determines the result of a subsequent measurement. While each atomin, say, the kth subensemble has a wave function of the form (1) with someunknown phases in the coefficients, we also know that it has the energyvalue Ek, because the energy has been measured and the atom has beenselected precisely on the basis of this energy value. But now, accordingto Born and Heisenberg, the transition probabilities |Snk|2 are independentof the actual wave function of the atom, so that if the atom is known tohave the energy Ek, the statistical distribution of the energy values uponrepetition of the measurement is simply given by (6) with cm(0) = δmk.If the repetition takes place immediately after the first measurement, thetransition probabilities |Snk|2 will tend to δnk, so that indeed the first resultwill be confirmed (Bacciagaluppi and Valentini, 2008, p. 175–176).

One might dispute that the description of measurements as randomising thephases should be taken literally: it might be simply a rather sloppy way oftalking about the decoherence induced by the measurement (encounteredsometimes even today in disussions of decoherence in general).18 However,the fact that Born and Heisenberg during the conference (and Heisenbergin the uncertainty paper) appear to use both the description of measure-ments in terms of phase randomisation and that in terms of reduction of thewave packet as equally good alternatives, may indicate that neither shouldbe taken literally. The wave function can be chosen one way or another,depending on what is more convenient ‘for practical purposes’.

3.2 Born’s discussion of the cloud chamber

In his discussion of the cloud chamber, Born attributes to Einstein the ques-tion of how one can account for the approximately straight particle trackrevealed by a cloud chamber, even if the emission of an α-particle is undi-

18My thanks to Antony Valentini for pointing out that a description of measurementin terms of phase randomisation appears also in Bohm’s textbook on quantum mechanics(Bohm, 1951, pp. 122, 600–602).

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rected, so that the emitted wave function is approximately spherical.19 Bornasserts that to answer it (p. 483):

.... one must appeal to the notion of ‘reduction of the probabilitypacket’ developed by Heisenberg. The description of the emissionby a spherical wave is valid only for as long as one does notobserve ionisation; as soon as such ionisation is shown by theappearance of cloud droplets, in order to describe what happensafterwards one must ‘reduce’ the wave packet in the immediatevicinity of the drops. One thus obtains a wave packet in the formof a ray, which corresponds to the corpuscular character of thephenomenon.

But then Born goes on to consider if wave packet reduction can be avoidedby treating the atoms of the cloud chamber, along with the α-particle, as asingle system described by quantum theory, a suggestion that he attributesto Pauli. The latter had made this suggestion also in a letter to Bohr oneweek before the beginning of the Solvay conference:20

This is precisely a point that was not quite satisfactory in Heisen-berg [(1927)]; there the ‘reduction of the packets’ seemed a littlemystical. Now in fact it should be stressed that such reductionsare not necessary in the first place if one includes in the sys-tem all means of measurement. But in order to describe at allobservational results theoretically, one has to ask what one cansay alone about a part of the whole system. And then from thecomplete solution one sees immediately that, in many cases (ofcourse not always), leaving out the means of observation can beformally replaced by such reductions.

Born’s own opinion is as follows (p. 483):

Mr Pauli has asked me if it is not possible to describe the processwithout the reduction of wave packets, by resorting to a multi-dimensional space whose number of dimensions is three times the

19Cf. Einstein’s main contribution to the general discussion (pp. 486–488), and above,section 2.1.

20Pauli to Bohr, 17 October 1927, in Pauli (1979, p. 411).

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number of all the particles present .... . This is in fact possibleand can even be represented in a very anschaulich manner [d’unemaniere fort intuitive] by means of an appropriate simplification,but this does not lead us further as regards the fundamentalquestions. Nevertheless, I should like to present this case here asan example of the multi-dimensional treatment of such problems.

Both Born and Pauli thus seem to think that the reduction of the wavepacket is a dispensable element in the description of measurements.21 How-ever, Born’s subsequent discussion remains somewhat unclear about whythis should be so. From the above quotation, it appears that the discussionis intended mainly as an illustration of the use of configuration-space wavefunctions (a point reiterated by Born at the end of his discussion). Born,indeed, merely presents a multi-dimensional treatment of the problem, sim-plified in that all motions are in one dimension and the cloud chamber isrepresented by only two atoms. Only in the end does Born remark that(p. 486):

To the ‘reduction’ of the wave packet corresponds the choice ofone of the two directions of propagation +x0 , −x0, which onemust take as soon as it is established that one of the two [atoms]1 and 2 is hit ....

Now, provided this remark is at all relevant to the question of whether wavepacket reduction is unnecessary, it should be read as an alternative to thedescription by means of reduction. That is, one should be able to leavethe wave packet uncollapsed and choose instead a direction of propagationfor the α-particle, either because this is truly what happens upon measure-ment, or because the two descriptions are equivalent at least ‘for all practicalpurposes’, in which case presumably neither is to be taken literally.

Incidentally, the atoms in the cloud chamber are described by Born on thesame footing as the α-particle, making this perhaps the first example ofexplicit inclusion of a measuring apparatus in the quantum mechanical de-scription. Note that the fact that the Schrodinger equation was not ap-plied to the measurement interaction means that there was no awareness at

21Note that also Pauli’s remarks to Heisenberg about transition probabilities and Bornand Heisenberg’s treatment thereof, discussed in section 2.2, crucially make reference to‘what one can say alone about a part of the whole system’. Pauli’s suggestion to Born andhis remarks to Heisenberg may in fact be related.

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the 1927 Solvay conference of the ‘measurement problem’, in the sense ofmacroscopic superpositions arising from the measurement interaction. Forinstance, also in Bohr’s famous exchanges with Einstein between the sessionsof the conference (Bohr, 1949), Bohr applies only the uncertainty principleto the apparatus, and certainly not the Schrodinger equation, so that nomacroscopic superpositions are considered. As regards Born’s example ofthe cloud chamber, it could have been used in principle to raise this prob-lem. However, if the reading of Born and Heisenberg’s position suggestedhere is correct, it is not surprising that Born did not see the resulting macro-scopic superposition as a problem, since the ‘state’ of the α-particle (underthe given conditions) would correspond indeed to its direction of motion.

3.3 The exchange between Heisenberg and Dirac

Born’s remarks on the collapse of the wave function should be contrastedwith Dirac’s remarks on the same topic, also in the general discussion(pp. 494–495):

According to quantum mechanics the state of the world atany time is describable by a wave function ψ, which normallyvaries according to a causal law, so that its initial value deter-mines its value at any later time. It may however happen thatat a certain time t1, ψ can be expanded in the form

ψ =∑

n

cnψn ,

where the ψn’s are wave functions of such a nature that theycannot interfere with one another at any time subsequent to t1.If such is the case, then the world at times later than t1 will bedescribed not by ψ but by one of the ψn’s. The particular ψn

that it shall be must be regarded as chosen by nature.

This, according to Dirac (p. 495) is ‘an irrevocable choice of nature, whichmust affect the whole of the future course of events’. Dirac thus appearsboth to take the wave function to be a real physical object, and to take thecollapse of the wave function to be a real physical process, connected withlack of interference (an interesting point both from today’s perspective andfor the exchange with Heisenberg). But Dirac goes further, and recognises

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that there are circumstances where the choice made by nature cannot haveoccurred at the point where it might have been expected. Dirac considers atsome length the specific example of the scattering of an electron, concludingwith the following observation (pp. 495–496):

If, now, one arranged a mirror to reflect the electron wavescattered in one direction d1 so as to make it interfere with theelectron wave scattered in another direction d2, one would notbe able to distinguish between the case when the electron is scat-tered in the direction d2 and when it is scattered in the directiond1 and reflected back into d2. One would then not be able totrace back the chain of causal events so far, and one would notbe able to say that nature had chosen a direction as soon as thecollision occurred, but only [that] at a later time nature chosewhere the electron should appear. The interference between theψn’s compels nature to postpone her choice.

In Dirac’s manuscript of this discussion contribution,22 a cancelled versionof the last sentence begins with ‘Thus a possibility of interference ....’, whileanother cancelled version begins with ‘Thus the existence of interference ....’.Possibly, Dirac hesitated here because he saw that in principle the mirrorcould always be added by the experimenter after the scattering had takenplace. Thus, there would be no cases in which interference could be ruledout as impossible, making this an unrealisable criterion for the occurrenceof collapse.

Precisely this point was made by Heisenberg, shortly afterwards in the dis-cussion (p. 497):

I do not agree with Mr Dirac when he says that, in the de-scribed experiment, nature makes a choice. Even if you placeyourself very far away from your scattering material, and if youmeasure after a very long time, you are ablef to obtain inter-ference by taking two mirrors. If nature had made a choice, itwould be difficult to imagine how the interference is produced. Ishould rather say, as I did in my last paper [(Heisenberg, 1927)],that the observer himself makes the choice, because it is only atthe moment when the observation is made that the ‘choice’ has

22AHQP-36, section 10.

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become a physical reality and that the phase relationship in thewaves, the power of interference, is destroyed.

Note the striking resemblance between what is said here by Heisenberg andwhat is said (more understatedly) by Born in his treatment of the cloudchamber. Born talks about the ‘choice of one of the two directions of prop-agation’, a choice which is taken not when one of the two atoms is hit, butwhen it is ‘established’ that it is hit (when the ionisation is ‘shown’ by theappearance of the cloud droplets); Heisenberg (who of course is also follow-ing Dirac’s terminology) talks of a ‘choice’ of which path is taken by theelectron, a choice which becomes physically real ‘only at the moment whenthe observation is made’. But Heisenberg goes further than Born here, sug-gesting that what happens upon observation is that ‘the phase relationshipin the waves, the power of interference, is destroyed’, i.e. that the effect ofmeasurement is phase randomisation rather than collapse.

4 Born and Heisenberg on ‘hidden variables’

To conclude, we shall now have a brief look at the views on what one wouldnow call ‘hidden variables’ (in particular in the context of guiding fields)expressed at the time by Born and by Heisenberg, mostly before the Solvayconference. Indeed, the idea of observables having values that are not strictlylinked to the wave function of the system (no ‘eigenstate-eigenvalue link’)might strike one as typical of hidden variables theories. This is preciselywhat happens in pilot-wave theories of the Bell type, as mentioned in sec-tion 3.1 above. Unsurprisingly, however, the views on the subject expressedby Born and by Heisenberg are quite negative.

4.1 Born on the practical irrelevance of microcoordinates

Consider Born’s second paper on collisions (Born, 1926b). In this paperBorn makes an explicit link between his work and guiding-field ideas, sayingthat while in the context of optics one ought to wait until the development ofa proper quantum electrodynamics, in the context of the quantum mechanicsof material particles the guiding field idea could be applied already, usingthe de Broglie-Schrodinger waves as guiding fields; these, however, determinethe trajectories merely probabilistically (p. 804). In the concluding remarks

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of the paper, Born comments explicitly on whether this picture is to beregarded as fundamentally indeterministic (pp. 826–827):

In my preliminary communication [(Born, 1926a)] I laid veryparticular stress on this indeterminism, since it seems to me tocorrespond perfectly to the practice of the experimenter. But ofcourse it is open to anyone who will not rest content therewith toassume that there are further parameters not yet introduced inthe theory that determine an individual event. In classical me-chanics these are the ‘phases’ of the motion, e.g. the coordinatesof the particles at a certain instant. It seemed to me unlikely atfirst that one could freely include quantities in the new theorythat correspond to these phases; but Mr Frenkel23 has informedme that perhaps this in fact can be done. Be it as it may, thispossibility would change nothing in the practical indeterminismof collision processes, since indeed one cannot give the values ofthe phases; it must lead, besides, to the same formulas as the‘phaseless’ theory proposed here.

Thus, Born took it that a ‘completion’ of quantum mechanics through theintroduction of further parameters into the theory would have no practicalconsequences, an opinion echoed in Born and Heisenberg’s report immedi-ately after their introduction of transition probabilities (p. 422):

While the determinateness of an individual process is assumedby classical physics, practically in fact it plays no role, becausethe microcoordinates that determine exactly an atomic processcan never all be given; therefore by averaging they are eliminatedfrom the formulas, which thereby become statistical statements.It has become apparent that quantum mechanics represents amerging of mechanics and statistics, in which the unobservablemicrocoordinates are eliminated.

At the Solvay conference the idea of quantum mechanics as eliminating mi-croscopic coordinates from the description of motions is mentioned by Bornalso in discussing Schrodinger’s treatment of the Compton effect (p. 371; cf.

23This is presumably Y. I. Frenkel, who at the time was in Germany on a Rockefellerscholarship. Born had supported Frenkel’s application. (See Frenkel, 1996, p. 72).

22

also p. 444). It may have been an important element of Born’s intuition,and appears also in Born’s reaction to the EPR paper (Einstein, Podolskyand Rosen, 1935).24

4.2 Heisenberg and Einstein on hidden variables

The above statements by Born may not rule out unequivocally the possibilityof thinking of the wave function as a guiding field (more so perhaps hisstatements in the adiabatic paper on the Unanschaulichkeit of the quantumjump). Heisenberg’s statements on the subject instead indicate both thathe understood the principles behind pilot-wave theories and that he rejectedthem decidedly.

Heisenberg’s views are contained in a letter to Einstein about the latter’sown unpublished hidden-variables proposal (cf. Pais, 1982, p. 444). In May1927, Einstein had proposed what in retrospect appears to be an alternativeversion of pilot-wave theory, with particle trajectories determined by themany-body wave function, but in a way different from that of de Broglie’stheory. This theory was described in a paper entitled ‘Does Schrodinger’swave mechanics determine the motion of a system completely or only inthe sense of statistics?’,25 which was presented on 5 May 1927 at a meetingof the Prussian Academy of Sciences. On the same day Einstein wrote toEhrenfest that ‘.... in a completely unambiguous way, one can associate def-inite movements with the solutions [of the Schrodinger equation]’ (quotedin Howard, 1990, p. 89). However, on 21 May, before the paper appeared inprint, Einstein withdrew it from publication. The paper remained unpub-lished, but its contents are nevertheless known from the manuscript versionin the Einstein archive — see also Belousek (1996) and Holland (2005).

Heisenberg had heard about Einstein’s theory through Born and Jordan,and on 19 May — just two days before Einstein withdrew the paper —wrote to Einstein enquiring about it. On 10 June 1927, Heisenberg wroteto Einstein again, this time with detailed comments and arguments againstwhat Einstein was (or had been) proposing. I shall now briefly summarisethis second letter.26

24See Born to Schrodinger, 28 June 1935, AHQP-92, section 2 (in German).25‘Bestimmt Schrodingers Wellenmechanik die Bewegung des Systems vollstandig oder

nur im Sinne der Statistik?’, Albert Einstein Archive 2-100.00; currently available on-lineat http://www.alberteinstein.info/db/ViewDetails.do?DocumentID=34338 .

26Heisenberg to Einstein, 19 May and 10 June 1927, Albert Einstein Archive 12-173.00

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Evidently, Einstein had not sent the withdrawn paper in reply to the originalenquiry, for Heisenberg mentions he has learnt nothing new, but Heisenbergsays he would like to write again why he believes indeterminism is ‘necessary,not just consistently possible’. If he has understood his viewpoint correctly,Einstein thinks that, while all experiments will agree with the statisticalquantum theory, nevertheless in the future one will be able to talk alsoabout definite particle trajectories. Heisenberg’s main objection is now asfollows.

Consider free electrons with a constant and low velocity, ‘so slow, that thede Broglie wavelength is very large compared to the size of the particle, i.e.the force fields of the particle should be practically zero on distances of theorder of the de Broglie wavelength’. Such electrons strike a grating withspacing comparable to their de Broglie wavelength. Heisenberg remarksthat, in Einstein’s theory, the electrons will be scattered in discrete spatialdirections. Now, if the initial position of a particle were known one couldcalculate where the particle will hit the grating and ‘set up some obstaclethat reflects the particle in some arbitrary direction, quite independently ofthe other parts of the grating’. This could be done, if the forces between theparticle and the obstacle act indeed only at short range, small with respectto the spacing of the grating. Heisenberg then continues:

In reality the electron is reflected independently of the obstaclein question in the definite discrete directions. One could onlyescape this if one sets the motion of the particle again in directrelation to the behaviour of the waves. But this means that oneassumes that the size of the particle, that is, its interaction forces,depend on the velocity. Thereby one actually gives up the word‘particle’ and loses in my opinion the understanding for why inthe Schrodinger equation or in the matrix Hamiltonian functionalways appears the simple potential energy e2/r. If you use theword ‘particle’ so liberally, I take it to be very well possible thatone can define also particle trajectories. But the great simplicitythat in the statistical quantum theory consists in that the motionof the particles takes place classically, insofar as one can talk ofmotion at all, in my opinion is lost.

Heisenberg then notes that Einstein seems willing to sacrifice this simplicity

and 12-174.00 (both in German). Passages from the letter of 10 June are quoted with thekind permission of Prof. Helmut Rechenberg of the Werner Heisenberg Archive.

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for the sake of maintaining causality. However, even Einstein’s approachwould not be able to change the fact that many experiments would be de-termined only statistically: ‘Rather we could only console ourselves withthe fact that, while for us because of the uncertainty relation p1q1 ∼ h theprinciple of causality would be meaningless, the good Lord in fact wouldknow in addition the position of the particle and thereby could preserve thevalidity of the causal law’. Heisenberg concludes the objection by sayingthat he finds it ‘actually not attractive [eigentlich doch nicht schon] to wantto describe physically more than the connection between experiments’.

Note that Heisenberg’s objection is not that the theory does not predictthe usual scattering pattern in the practically unrealisable case in whichone manipulates the trajectory of a particle with known initial position.Rather, his gedankenexperiment serves to establish the point that, even inthe normal case (in which the initial position of the particle is unknown),the direction in which a ‘particle’ is scattered must depend only on the localfeatures of the grating, thus contradicting the normal experimental results.The only way to have the direction of scattering depend on the features ofthe grating other than where the particle hits it, is to make the trajectoryof the particle depend on the associated wave rather than on particle-likeshort-range interaction behaviour.

It is striking that Heisenberg’s objection concerning the electron and thegrating shows that he thought that a trajectory-based deterministic theoryof quantum phenomena is possible. It is equally striking that Heisenbergappears to have thought that such a theory is nevertheless unacceptable onwhat would seem to be aesthetic grounds (or grounds of Anschaulichkeit),because it gives up both the usual concept of particle and the mathematicalsimplicity of quantum mechanics. This objection appears to have remained amainstay of Heisenberg’s negative views on hidden variables. Indeed, Heisen-berg repeated it also in his own draft reply to the EPR paper (Heisenberg,1985, p. 416).27

27My thanks to Elise Crull for directing my attention to this passage in Heisenberg’sdraft.

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5 Conclusion

In this paper, I have suggested that Born and Heisenberg’s report at the1927 Solvay conference is significant because it presents a more mature anddefinitive version of the statistical interpretation of quantum mechanics.The key point about this suggestion is that the interpretation in the reportmerges elements of Born’s interpretational work of 1926 and of Heisenberg’swork on fluctuations and in the uncertainty paper. I have also proposeda specific reading of Born and Heisenberg’s position (thereby continuingwhere the analysis of Bacciagaluppi and Valentini, 2008, leaves off). Thekey intuition behind this proposal is that Born and Heisenberg did not takethe wave function to be a real entity.

Of course, it is well-known that Heisenberg at least was strongly antagonis-tic to Schrodinger’s introduction of wave functions and to his attempts tointerpret them as giving an anschaulich picture of quantum systems. WhileBorn’s work of 1926 can be put in relation with ideas on guiding fields, Isuggest that, at least come 1927, Born and Heisenberg’s conception of thewave function was thoroughly statistical, i.e. more analogous to a classicalLiouville distribution, thus making also the collapse of the wave function amatter of convenience of description.

Born and Heisenberg’s own words give the impression that they consideredthe presentation in their report to be indeed a final formulation of the theoryand interpretation of quantum mechanics (pp. 409, 437):28

Quantum mechanics is meant as a theory that is in this sense an-schaulich and complete for the micromechanical processes ([Hei-senberg, 1927]) .... There seems thus to be no empirical argumentagainst accepting fundamental indeterminism for the microcosm.

.... we consider [quantum mechanics] to be a closed theory[geschlossene Theorie], whose fundamental physical and math-ematical assumptions are no longer susceptible of any modifica-tion.

Even as these views were being expressed, there remained significant dif-28One can recognise Heisenberg’s pen in these passages, which were in fact drafted by

him (Born to Lorentz, loc. cit., note 3). Cf. also Heisenberg’s later writings on the conceptof ‘closed theories’, e.g. Heisenberg (1948).

26

ferences of opinion even within the ‘Gottingen-Copenhagen’ camp (as seenin the exchange between Dirac and Heisenberg). Moreover, with its lack ofcollapse and perhaps even of fundamental wave functions, the interpretationpresented was itself quite different from what might be assumed today tohave been the ‘statistical interpretation’ of quantum theory.

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